Question: Find the only positive real number $x$ for which $\displaystyle \frac{x-4}{9} = \frac{4}{x-9}$.
Explanation: The first approach that comes to mind is probably also the best one.  So we cross-multiply to obtain $(x-4)(x-9) = 36$.  Multiplying out the left-hand side and cancelling the 36 yields $x^2-13x = 0$, or $x(x-13)=0$.  This equation has two solutions, $x=0$ and 13.  Since we are looking for positive answers, we take $x=\boxed{13}$.